/sci/ - Science & Math

>>11591121
Why would the first and third be true?

They are all true though

>>11591121
All of them are false. Anything else is Jewish mysticism.

>>11591136
cringe and bluepilled
>>11591194
based and redpilled

>>11591121
>1 = 1
>1 = 2
>1 = 3
>why is only one of these statements true ?

>>11591450
Trick question. They're all true depending on which axioms you choose to use.

>>11591121
.88.... = 8/9 =/= .9
.99... = ?/? ?= 1
1.0... = 1/1 =/= 1.1

.9... doesn't exist

>>11591121
Read.

>>11591486
Bluae.

>>11591194
low IQ and angry he is not a chosen person

>>11591121
There are numbers between 0.888... and 0.9 and between 1.000... and 1.1:
0.88... < 0.89 < 0.9
1.00... < 1.01 < 1.1
thus they are not equal. there is no such number between 0.999... and 1.0 (provide a counterexample if I'm wrong) because they are equal

>>11591121
Whatta fool we have here

>>11591121
8/9 = 9/10
9/9 = 10/10
1 = 11/10
Why is only one of these statements true?

>>11591121
Because:
x=0.999...
10x=9.999...
10x=9+x
9x=9
x=1

This does not work for the others.
>>11591519
kill yourself Jew.

>>11591450
False equivalence. Doesn't fit ops pattern. >>11591486
Dodging a simple question.
>>11591645
Counterexample 0.999...95
>>11591810
You can't just say this does not work for the others. You have to prove somehow that there can be no way to use made up algebraic manipulations to get the others like you did for 0.999

>>11591804
False equivalence.

>>11591887
>0.999...95
is less than 0.999...

>>11591899
Brainlet

>>11591887
>made up algebraic manipulations
Where exactly did I use a made up algebraic manipulation? All of those are legal. I set 0.999... to a variable, multiplied it by 10, realized that 10*x is the same as 9+x (if you truly can't see that this isn't subtraction but rather just an equivalent statement then you're braindead), then subtracted x from both sides and divide by 9. Care to show me how exactly I used made up algebra?

>>11591892
They're literally the same as the equations in OP's pic, except the numbers are written as fractions instead of decimals.

>>11591913
Those fractions are not equal to any of those decimals, that's the point

>>11591906
You're doing the same shit as the numberphile proof of - 1/12. It's made up shit

>>11591917
You don't think 0.888... = 8/9?

>>11591921
I don't watch numberphile because I'm not a retarded pseud. My proof is valid and you cannot find any false algebraic manipulations in it. If I say 0.999...=1, and can prove that fact with math, then it is true. Maybe because you are a stupid brainlet and don't understand how numbers work this is difficult for you to understand.

>>11591922
No

>>11591121
fuck ponds
fuck girls
fuck needles
Why is only one of these statements true?

>>11591887
>0.999...95
at which position is the five ?

>>11591941
Why not?
https://www.wolframalpha.com/input/?i=8%2F9

>>11591941
So what happens when you do long division of 8 by 9?

>>11591941
When did you receive brain damage?

>>11591899
>a number with finite decimal places is greater than a number with infinite decimal places
Ah yes I see

>>11591481
>.9... doesn't exist
Fuck off you mathematically illiterate retard.

>>11591902
>a number with finite decimal places is greater than a number with infinite decimal places
Ah yes I see

>>11591935
This is what you are doing

1-1+1-1+1-1+...=(1-1)+(1-1)+(1-1)+...=0
1-1+1-1+1-1+...=1-(1-1)-(1-1)-...=1
0=1

It's made up shit because you close your eyes and wish infinity works the way you want it to

>>11591947
Omegath

>>11591902
Clarification: are you calling the ".9...<.9...95" guy the brainlet for being mathematically illiterate, or the responder for replying to bait?

>>11591950
A calculator just gives you an increasingly accurate approximation. It never reaches the true value

>>11591981
Stop being a retard who doesn't understand how decimals work.

>>11592002
So you don't have any argument and just attack me? The delusions of onetards...

>>11591981
You cannot name a digit (e.g. tenths digit, hundredths digit) which is not 8 when calculating the decimal approximation of 8/9 to some arbitrarily high precision.

>>11592010
The arguments have already been made. Decimal representations are not like a train that chugs along. The numbers are ALREADY there, ALREADY contributing to the value. "Approach" has an entirely different meaning in Math than it does in common vernacular. By denying this, and arguing that the decimal representation "never reaches" the true value, you are just like the creationists who stupidly argue that evolution is "just a theory". I repeat, stop being a retard who doesn't understand how decimals work.

>>11592025
When you plug in into wolfram alpha the computer does a series of finite steps dumbass. That's why you can't use a calculator to prove the equality, because they're different

>>11592035
>A computer can't do it, therefore it's impossible.
Fuck you, just because YOU can't do math better than a computer doesn't mean I am similarly retarded. The numbers ARE THERE, the fact that a computer can't show all of them is irrelevant, retard.

>>11592023
The limit does not conserve equality. You know that troll picture that says pi=4? That is you right now. You are being a troll

>>11592040
LOL smoothbrain. You literally cannot understand the concept of not rounding

>>11592042
>>11592045
>>11591971
What number are you referring to when you use the notation "0.999..."? You can use the following ways to describe the number you are referring to: the supremum of a bounded set of rational numbers, a unique zero of a continuous function, the limit of a Cauchy sequence. If you cannot use any of these to uniquely identify what number you are referring to by "0.999...", then it is not a real number. If you disagree that you are required to use the above ways to describe this number, then the conversation should be shifted to the merits of the definition of real numbers.

>>11592073
We are referring to one. Proof:
>>11591966

>>11592045
Don't accuse people of what you are guilty of.

>>11592042
An infinitely crinkled square crinkled up to look like a circle is not a circle. The infinite sum 8/10 + 8/100 + 8/1000 + ... actually is 0.888...

>>11592117
oooooooooooooooooooh genius

>>11591121
0.888...888 = 0.888...890 = 0.888...900 = ... = 0.890... = 0.900...
seems right to me

>>11591121
PhD here, to no surprise sci is not able to grasp basic calculus, despite also being a math board. The reason behind them all being true is quite trivial, and I hope that most in this thread are, 'baiting'. Now, for the first one, one who cannot grasp the nature of infinity, may say that no matter how many decimal places, you will never reach 0.9. Now, this is correct, but there is an infinite amount of 8s and we know that as you add more and more 8s you approach 0.9, so, the notation 0.888.. is equivalent to 0.9. I don't have to talk much about 0.9999..., use the same analogy and the reason will come to you trivially. Now, let us assume 1.000.. Is equal to 1, now, if we multiply it by infinity we get infinity, but shouldn't it be infinity + 1? You may think that it does not make a difference but it does, lim(h->Inf) (h+1)-h = 1. 1.0000.. Must be equal to 1.1, when it is multiplied by infinity, this happens, infinity(1 + 0.1), as you can see, 0.1 gets multiplied by infinity, this is the same as it being devided by 1/infinity, and this means that if it is equal to 1.1 when multiplied by infinity it is infinity + 1. Thank you for reading this far, please make sure you are informed before making such stupid arguments.

>>11592166
Replace all the '=' with '<' and you'd be correct.

>>11592203
You are an inspiration to all. You are proof that even a mathematically illiterate, literal retard can get a PhD.

>>11592216
wrong. surely you learned how decimals work in school

>>11592251
I did, and clearly you didn't.

>>11592099
I don't think a proof like this would convince anyone in the 0.999... =/= 1 camp, since I suspect there are definitional disagreements. Which is why I suggested shifting the conversation to the how the real numbers are defined, and why its notational conventions formed the way they did.

>>11592274
Of course it wouldn't convince them, understanding the proof requires knowledge of basic analysis.

>>11591121
None of those are true.
Why would two different numbers be the same?

>>11592203
Can someone help a mathlet out? (A) why h for a variable there? (B) why would zero count as a “real” quantity (obviously it’s a digit but it has no value, right?)

>>11592313
The second equality is true. See below:
>>11591966

>>11592328
I have no idea what any of that means.
But the way I see it, numbers exist to represent concepts, such as a singular whole, aka 1.
By that definition, 0.999... represents a number that infinitely continues to creep closer to 1 while at the same time never becoming 1.
They're different concepts used for different purposes, I don't see how they could be considered the same.

>>11592350
They are considered the same for the same reason 9/9 is the same as 1. It's a different way of writing the same number.
Also the idea of "becoming" is not valid here. The nines are already there, and always contributing to the value of point nine repeating, aka one.
Not a proof, but here's another way to look at it:
1/9=.111...
2/9=.222...
...
8/9=.888...
9/9=.999...=1

>>11591121
Second one.

x = 0.999...
10x = 9.999...
9 + 0.999... = 9.999...
9 + x = 10x
10x - x = 9
9x = 9
x = 1
>>11592350
>0.999... represents a number that infinitely continues to creep closer to 1
A number doesn't "creep" anywhere.

>>11591121
0.777 = 1
0.666 = 1
0.555 = 1
0.444 = 0
0.333 = 0
0.222 = 0
0.111 = 0

>>11592368
I've been trying to argue against this for 40 minutes only to come to the conclusion that I was wrong.
0.111... and 1/9 ARE actually used for the same purpose, to represent the concept of one out of nine.
The reason we use 0.111... is because of limitations in our numbering system.
So, fair enough, I admit that 0.999...=1.

>>11592549
Yay! Welcome to the team! *high five*

>>11591971
No, that's not what I'm doing. Please point out where I did anything close to that. I subtracted, made equivalent statements, multiplied, etc. Everything in what I provided stands completely true. You still have not pointed out here I made a mistake or a false algebraic manipulation, you've just pointed out what completely unrelated people have done.

>>11591887
lmao retard

>>11592350
>I have no idea what any of that means.
I'll explain
real numbers are constructed from the rationals in an analogous way to which the rationals (that is, numbers that can be represented as a fraction) are constructed from the integers. To get an idea of how the rationals can be constructed, realize that every rational rational can be expressed as a/b where a is an integer and b is a natural number. All the properties of rational numbers can be reduced to the properties of a and b. For example, we say that two rational numbers a/b and c/d are equal if and only if a * d = b * c. Notice that a,b,c and d are all integers or natural numbers, and yet we have expressed the condition for equality of two non-natural numbers a/b and c/d. To be clear, we say two rationals a/b and c/d are equal if and only if a*d = c*b. That is what equality means for rationals.

The reals can be constructed in a similar way, two commonly ways commonly used to construct the reals are 'Dedekind cuts' and 'Cauchy sequences'. These are equivalent ways to construct the reals from the rationals, and what it means for real numbers to be equal can be precisely defined in a way that is analogous to the way we can precisely define what it means in terms of the rationals that can be used to define each real number. If we use Cauchy sequences to construct the reals, we can define the reals by an infinite sequence of rationals, and we say two reals are equal if and only if for any positive number [math]\epsilon[/math], however small just not equal to 0, we can always go to a term far enough along in the cauchy series defining each real that the difference between those rationals after that point in the sequence is always smaller than [math]\epsilon[/math]. That defines equality of the real numbers, and if you think about the sequences that you could use to define 0.999... (0.9, 0.99, 0.999, and so on) and 1 (1, 1, 1, and so on) you can always do that. That is what equality means for reals.

>>11592630
>I have no idea what any of that means.
I'll explain
real numbers are constructed from the rationals in an analogous way to which the rationals (that is, numbers that can be represented as a fraction) are constructed from the integers. To get an idea of how the rationals can be constructed, realize that every rational rational can be expressed as a/b where a is an integer and b is a natural number. All the properties of rational numbers can be reduced to the properties of a and b. For example, we say that two rational numbers a/b and c/d are equal if and only if a * d = b * c. Notice that a,b,c and d are all integers or natural numbers, and yet we have expressed the condition for equality of two non-natural numbers a/b and c/d. To be clear, we say two rationals a/b and c/d are equal if and only if a*d = c*b. That is what equality means for rationals.

The reals can be constructed in a similar way. Two commonly used ways are 'Dedekind cuts' and 'Cauchy sequences'. These are equivalent ways to construct the reals from the rationals; what it means for real numbers to be equal can then be precisely defined in terms of rational numbers in a way that is analogous how we can define equality for rationals in terms of integers. If we use Cauchy sequences to construct the reals, we define the reals by an infinite sequence of rationals, and we say two reals are equal if and only if for any positive number ϵ, however small just not equal to 0, we can always go to a term far enough along in the two cauchy sequences that define each real number that the difference between the rational numbers after that point in the sequence is always smaller than ϵ. That defines equality of the real numbers, and if you think about the sequences that you could use to define 0.999... (0.9, 0.99, 0.999, and so on) and 1 (1, 1, 1, and so on) if you think about it, you'll realise you can always do that. That is what equality means for reals.

>>11592652
meant for >>11592350

>>11592615
That's what you're doing. I could say 1=0 because of the associativity of addition. But it's not when you deal with infinities. In the same way you are wrong

>>11592628
lel peabrain

>>11592471
Your ""proof"" is like the proof that 1+2+3+...=-1/12. Its mindless algebraic nonsense

>>11591121
lets start at 0.999...
0.999... is always .1 (of any denomination) away from being 1.
now, if you apply the same thing to 0.888..., you can add any denomination of .1 to any area of the number. for conveniences sake lets put the .1 in the thousandths place. so we'll make the .1 into .001 and put it into .888. all it does is become .889.
1.000... is literally just 1. 0.000...=0 no matter what. 1=1. 1+0=0.

>>11591887
>dodging a simple question
no the question is literally answered by the book in chapters 1-3. There's a reason why this seemingly untrue fact is true.

>>11591810
>pic related: a pretty corpse in nuremberg

>>11592980
it's not meaningless
I don't expect you to understand the proof without having studied real analysis, but I explained it in layman's terms here >>11592652.
Please tell me if there's anything you don't understand and I'll try to explain it in other terms.

Here is a proof to btfo 1tards with. Please copy and paste this to btfo 1tards in all threads

1-0.999...=0.00000000...000001

There are infinite 0s between the start of the number and the 1 at the end.
>hurr durr you cant do that
Well I guess you are saying infinity doesn't exist then? Because if there are infinite .9s before you get to 1 why cant you have infinite 0s before you get to .0000000000001?

1tards BTFO FOREVER!

>>11592992
Why do you have to invoke chapters of a textbook to answer a simple question? You're like a theologian who has dedicated his life explaining away contradictions instead of facing the obvious. You're God doesn't exist

>>11593057
Nice

>>11593057
Based

if 8/9 is 0.888
then by that logic 9/9 = 0.999
but 9/9 is also 1
9/9 = 0.999 = 1

>>11593074
circular reasoning

>>11593057
1) See >>11591966 for why it actually does equal one.
2) There can't be infinite zeroes inbetween the decimal point and one. You can't put a one at the END of an infinite series of zeroes because there isn't an END to the zeroes, retard.
3) Stop using phrases such as "before you get to 1" the nines are already there. There is no process of adding nines. Also, even if it was a process, the nines would work but your retarded "0 at the end" wouldn't because there isn't an end to the addition of nines, but there IS an end to your addition of zeroes.
4) NO UR 99.999...% BTFO 4EVER!

>>11593081
You're right, look at >>11591966 instead. That's an actual proof.

>>11593074
8/9 is not 0.888... tardo

>>11593084
There are infinite numbers in [0,1) and 1 comes after so obviously 1 can come after infinite zeros

>>11591221
>>11591902
>>11591941
>>11592045
>>11592975
>>11593069
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>>11593084
>2) There can't be infinite zeroes inbetween the decimal point and one. You can't put a one at the END of an infinite series of zeroes because there isn't an END to the zeroes, retard.
Uh, you don't look like the almighty Jesus to me and you don't get to make those rules, sunshine. If you can have infinite 0s after a 1 or infinite 9s as a 1 I see no reason you couldn't put infinite 0s before a 1. It seems you mathematical brainboxes drink your own kool-aid too much. It seems to me you are limited in your thinking to differentiate the "start" from the "end" of a number, and I see no reason the infinite portion of a decimal can't come before the last digit. If an infinite thing can have a begging, it can surely have an end, or maybe those things are just the same

>>11593111
wasted trips

>>11593111
checked
based

>>11593103
Yes it does. Literal grade schoolers learn this stuff ffs.

>>11593119
It's just really close it will never equal 8/9

>>11593115
If you read and understood 1), then you'd know that we didn't "just decide" that .9...=1 we PROVED IT based on the definitions of numbers and operations involved. Your mind isn't open, it's empty.

>>11591121
0.888... = 0.89

>>11593111
OOOOOHOOOH OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HU HEEEE HAHAHA AH A AAAAAAH AAAAAAHOOOOOHOOOH OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HUHEEEE HAHAHA AAAAH A AAAAAAHAAAH A AAAAAAH AAAAAAHOOOOOHOOOH OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HU OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HU HEEEE HAHAHA AH A AAAAAAH AAAAAAHOOOOOHOOOH OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HEEEE HAHAHA AAAAH A AAAAAAHAAAH A AAAAAAH AAAAAAHOOOOOHOOOH OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HU OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HU HEEEE HAHAHA AH A AAAAAAH AAAAAAHOOOOOHOOOO OOH HUHU H UHU UH HU OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HU HEEEE HAHAHA AH A AAAAAAH AAAAAAHOOOOOHOOOH OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HEEEE HAHAHA AAAAH A AAAAAAHAAAH A AAAAAAH AAAAAAHOOOOOHOOOH OOH H OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU HUHAHAHA AH A AAAAAAH AAAAAAHOOOOOHOOOH OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HEEEE HAHAHA AAAAH A AAAAAAHAAAH A AAAAAAH AAAAAAHOOOOOHOOOH OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HU OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HU HEEEE HAHAHA AH A AAAAAAH AAAAAAHOOOOOHOOOO OOH HUHU H UHU UH HU OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HU HHO OOH HUHU H UHU UH HU HEEEE HAHAHA AH A AAAAAAH AAAAAAHOOOOOHOOOO OOH HUHU H UHU UH HU OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HU HEEEE HAHAHA AH A AAAAAAH AAAAAAHOOOOOHOOOH OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HEEEE HAHAHA AAAAH A AAAAAAHAAAH A AAAAAAH AAAAAAHOOOOOHOOOH OOH H OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU HUHAHAHA AH A AAAAAAH AAAAAAHOOOOOHOOOH OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HEEEE HAHAHA AAAAH A AAAAAAHAAAH A AAAAAAH AAAAAAHOOOOOHOOOH OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HU OOH OOH HAHAHA HAA... OOHO OEEEE HAHAHA AH A AAAAAAH AAAAAAHOOOOOHOOOH OOH OOH HAHAHA HAA... OOAH A AAAAAAH AAAAAAHOOOOOHOOOH OOH OOH HAHAHA HAA... OOHO OOH HUHU H UHU UH HEEEE HAHAHA AAAAH A AAAAAAHAAAH A AAAAAAH AAAAAAHOOOOOHOOOH

>>11593111
C R I N G E

>>11593110
Complete non-sequitur.

>>11593130
proved it based on your own mathematical rules you all wank over when they could be WRONG. Only GOD and JESUS know the true nature of the universe, not YOU.

>>11593133
8/9 =/= 89/100
Fucking retard.

>>11591966
.999... + .999... = 1.999..8

>NOOOOOOOOOOOO YOU CAN'T PUT AN 8 AT THE END, .999 IS INFINITE AND THE 8 ISN'T FAST ENOUGH TO REACH THE END OF THE INFINITE NUMBER EVEN THOUGH THE NUMBER ADDED TO IT IS ALSO INFINITE!!!11!1

>>11593141
Oh fuck off Jesus-tard. You are the reason why everyone hates Christians.

>>11593146
>I can't understand the math involved so I'll use a wojak meme instead.
Fuck off.

>>11593151
Can I get a cope lads?

>>11593133
0.888... = 0.889

>>11591966
This proof is fallacious in that it assumes ϵ=0

>>11593146
>>11593158
The real numbers aren't fucking order-isomorphic to lexicographic order on [math]10^{\omega+1}[/math], brainlet.

>>11591966

>>11593057
>Because if there are infinite .9s before you get to 1 why cant you have infinite 0s before you get to .

Proposition: [math]1 - 0.\underbrace{999...}_\text{$\infty$ 9s}[/math] is not equal to [math]0.0\underbrace{0...0}_\text{$\infty$ 0s}01[/math].

Proof: Suppose the contrary. Then there must exist some last digit equal to 1 in the decimal expansion. The terminating digit must come after an infinite number of digits after the decimal place, as otherwise it would be equal to
[eqn] 1 - 0.\underbrace{999...9}_\text{$N$ 9s}[/eqn] where [math]N[/math] is a finite number, contrary to assumption. As it is impossible for a digit in a decimal expansion to come after an infinite number of digits after the decimal place (proven in the lemma) there can be no such digit. Therefore, the proposition is true.

Lemma: There can be no digit which comes after an infinite number of digits in a decimal expansion
Proof:
1. The first digit of a decimal after the decimal place has come after a finite number of digits after the decimal place (to be precise, 0)
2. Suppose it has been proven for the first [math]n[/math] digits, i.e. the number of digits after the decimal place it has come after, [math]n-1[/math], is finite. As the [math]n+1[/math]th digit comes immediately after the [math]n[/math]th digit, it must come after [math]n[/math] digits. As [math]n[/math] is 1 greater than [math]n-1[/math], and [math]n-1[/math] is finite, it follows that [math]n[/math] is also finite. Thus the [math]n+1[/math]th digit has come after a finite number of digits.

>>11593224
>this infinity is faster than that one

>>11593241
err what?

>>11593178
what? no it doesn't.

>>11593206
based

>>11592973
Where exactly did I use associativity of addition?

>>11593178
literally the proof does not do that

>>11593272
None of the algebra rules work with infinity my friend. Every step of your proof is wrong

>>11593206
>>11593271
Now look up Cauchy's [math] \delta [/math] - [math] \epsilon [/math] definition of limits.

>>11593224
Your proof fails because you assume the decimal places are countable

>>11593328
We are talking about numbers here. Remember numbers? Not this Greek symbol manipulation bullshit

>>11593342
They are. Be my guest: go ahead count and them!

>>11593328
It's literally the same definition

>>11593348
But [math] \delta [/math] and [math] \epsilon [/math] ARE real numbers.

>>11593366
You can count the reals too doesn't mean they are countable

>>11593372
The same as the one given in the proof that 0.999... = 1. I agree.

>>11593378
seethe

>>11593374
Uhhh hello? They are Greek letters. Apparently using the alphabet is not obfuscation enough for you tards

>>11593381
Seethe... about being right?

>>11593147
>taking the obvious bait
You’re as bad as the trolls.

>>11593390
Keep on wronging, wronger

>>11592980
So which line of the proof do you disagree with?

>>11593390
That you wasted time typing up that paper without checking if you actually knew what you did.

>>11593398
>>11593323

>>11593058
ok retard

>>11593397
But I'm not wrong, and no one has shown me to be wrong.

>>11593376
>You can count the reals too
you literally can't
regardless there is a bijective mapping from the natural numbers to digits after the decimal place of the decimal places of a decimal number [math].d_1d_2d_3...[/math] here it is:
[eqn]
f(n) = d_{n + 1}
[/eqn] thus the cardinality of the set of digits after the decimal of a decimal expansion is equal to the cardinality of natural numbers (i.e. is fucking countable)

>>11593141
DEUS VULT DEUS VULT

>>11593057
>>11593064
>>11593069
>samefagging yourself
Truly pathetic.

>>11593402
I didn't write it. I'm a different anon.

>>11591486
NO STOP TRYING TO CURE ME OF MY BRAINLETISM

>>11593147
>falling for entry-level bait
not very bright are ya

>>11593403
Just name a line.

>>11593408
You forgot to count d_infinity

>b-but there's no such thing!!!! Wtf?!!??!?!! My tiny brain cannot comprehend this!!!!!!!

So you're saying infinity doesn't exist?

>>11593432
Every line

>>11593438
d_infinity's non-existence can be easily be established by induction

>>11593449
Do it then

>>11593443
Ok, what’s wrong with the first line?
>Let x = 0.999...

>>11593453
It's proven by the lemma in my post here >>11593224

>>11593455
Every step between lines is wrong

>>11591941

>>11593462
>N is finite
>{0} is finite
>{0,1} is finite
>by induction {0,1,2,...} is finite

That's basically your lemma

>>11593483
well you missed out the inductive step that shows that if n is finite then n+1 is also finite, but yeah that's the jist of it.
what's your problem with that?

>>11593476
?

>>11593476
So 9x = 9 implies x = 1 is wrong?

>>11593476
so 10x - x is not 9x? you're a fucking retard

>>11593495
how the fuck can you subtract an infinite amount of 9s from a number? it's not possible, you couldn't do it in a finite amount of time

>>11593486
Infinity is not n plus anything. So your proof doesn't account for d_infinity. You're assuming that every decimal is d_n for natural n. It's circular reasoning

>>11593495
Exactly. Too hard for your tiny mind?

>>11593188
I too like my wormholes with a grain of quantum entanglement

>>11591906
x=0.955
10x=9.55
10x=9+x
9x=9
x=1

>>11593491
>>11593495
x = 0.999... does not imply 10x = 9.999... You can't just assume it works like that

>>11593531
Why.

>>11593536
if we re allowed to assume shit ill assume that the decimal representation of numbers is lexicographically ordered so 0.9999... < 1 qed

>>11591121
if you take the geometric series of 1/10 you get 10/9 if you subtract 1 you get 1/9 multiplying that by 9 gives you 1

>>11593530
Based dyslexic
9+x = 9+0.955 = 9.955 != 9.55

>>11593567
u passed the test

>0.999...

>>11592073
/thread

>>11593499
>how the fuck can you subtract an infinite amount of 9s from a number?
9.999... - 0.999... = 9

>it's not possible, you couldn't do it in a finite amount of time
I just did it in a finite amount of time.

>>11591887
x=0.888....
10x=8.888...
10x=8+x
9x=8
x=8/9=0.888

silly billy

>>11593118
samefagging this hard

>>11593531
>x = 0.999... does not imply 10x = 9.999...
Then what is 10x?

>>11593591

>>11591121
0.888... is equal to .9 if simplified
0.999... is equal to 1 if simplified
1.000... is equal to 1 simplified and unsimplified

>>11591975
This is not about ordinals you fucking retard. This is an endless stream of 9's. There is no 5, per the idea of 0.999...

>>11593531
0.9... aleph_0 9's after decimal point
9.9... aleph_0 9's after decimal point
yes you can, and everyone does

>>11593975
Wrong, .9...=1 exactly with or without simplification. See >>11591966

>>11593894
It cannot be simplified.

>>11594015
0.999... < 0.999...95 < 1
What is so hard to understand about this retard?

>>11594026
Everyone also assumes God is real. Wishful thinking doesn't make something real, sweetheart

>>11594068
You are the retard for thinking this means what you think it means. You are misusing notation.

>>11594073
Cool argument tard

>>11594068
>0.999...95
What do you think this notation means, anon?

A decimal sequence is a function from natural numbers to digits. The decimal sequence ".999..." corresponds to f(n) = 9. What function from natural numbers to digits are you talking about when you write ".999...95"?

>>11594095
f(n) = 9 if n < infinity, 5 otherwise

>>11594088
It actually is because in ".9..." the ellipses means "the nines continue forever". In the ".9...95" the ellipses mean "the nines continue for a finite amount of times until it ends with a nine and five". That is what the notation means, you are wrong if you disagree. Shame on you for being a retard while calling others retarded.

>>11594104
n is a natural number, so n < infinity is always true. Which means your ".999...95" means the same thing as ".999...", which means it is equal, not larger.

>>11594104
Which, and you'd know this if you weren't the retard in this situation, simplifies to f(n)=9.

>>11594110
n can be infinity. Close minded of you to think that it can't

200 replies, really?

>>11594121
No it can't. It's not a matter of closed mindedness, it's a matter of actually knowing the fucking definitions of the terms and math involved. Read a fucking analysis textbook ffs.

>>11594107
Wrong. 0.999...95 means there are infinite 9s in between.

Let me break it down since you are special. Consider the family of intervals (-n,n) together with R, where n is in N. Its are totally ordered wrt set inclusion, infinite, and has maximum element R. Each (-n, n) has finite length but the maximum element R is infinitely long. This proves that something different can come after infinitely many similar things

>>11594131
If you had read an analysis book you would know that the real number line can be extended to include positive and negative infinity instead of blindly mistrusting infinities and wanting to round them

>>11594138
But we are not talking about real numbers. As >>11594095 said, a decimal sequence is a function from NATURAL numbers to digits. Infinity is not a natural number. Yes, you can create an extension of the natural numbers that includes infinity (such as the ordinal numbers), but that is not what a decimal sequence is. A decimal sequence is a function from the standard, regular, un-extended natural numbers to digits, and infinity is not a natural number.

>>11594138
If YOU had read an analysis textbook, then >>11591966 would make sense to you, and you'd agree that .9...=1 and thus there's no such thing as .9...95.

>>11593178
you dumb motherfucker

>>11594071
[citation needed]

>>11593342
They are trivially countable. What the hell are you talking about you god damn retard?

>>11594192
I wish world peace is a thing that exists. It isn't, therefore wishful thinking doesn't make something real. QED

>>11591975
no such thing in real numberes

{1} is bounded
{1,2} is bounded
{1,2,3} is bounded
{1,2,3,4} is bounded
...
by induction N = {1,2,3,....} is bounded. Therefore there exists a largest natural number, which is necessarily the exact amount of 9's in 0.999... Otherwise the notation wouldn't make sense. Since there's only a finite number of 9's, clearly 0.999... < 1.

Am I doing it right boys ?

>>11594289
blue pilled response

>>11594154
Nope that's not how it works

>>11594287
You proved that the set [math]\lbrace n|n\in\mathbb{N}, n \leq N\rbrace[/math] is bounded for all [math]N\in\mathbb{N}[/math], which is of course true. You didn't prove that the natural numbers are bounded and it can be easily shown by contradiction that they are not:
If [math]\mathbb{N}[/math] is bounded then there exists a largest natural number Q. Its successor, Q+1 is also a natural number by Peano axiom 6. Thus Q+1 is a natural number. But Q+1>Q, contrary to assumption, thus [math]\mathbb{N}[/math] is unbounded.

>>11591947
the ...th you idiot

>>11592203
i seriously hope you are the one baiting pal.

>>11594742
+1

>>11594742
>...th
not a natural number

>>11593595
>9.999... - 0.999... = 9
false

>>11595219
[math]9.999... - 0.999... = 9 + \sum_{n=1}^{\infty}\frac{9}{10^n} - \sum_{n=1}^{\infty}\frac{9}{10^n} = 9[/math]
seems okay to me

>>11595227
you don't know what you're talking about baka

>>11595246
find the mistake

>>11595250
you can't subtract an infinite number of something.

>>11595258
I think I just did

>>11595267
you didn't

>>11595219
>>11595246
>>11595258
What do you gain from this?
Honestly. (You)s?
Attempt to prove that [math]9.999... \neq 9 + 0.999...[/math], and come back with your results.
Simply (You)ing a proof of the contrary and saying "false" does not prove your statement.

We have well-established rules to handle rational numbers and subtraction of those, that so far haven't lead to contradictions.
So start by proving that 0.999... is no rational number. That would be a good start, I suppose.
That may also unveil where your misunderstanding of 0.999... originates from.

>>11595268
I think you need better arguments than "you can't" and "you didn't"

>>11595286
You'd have to subtract an infinite number of numbers. Have you ever seen an infinite number of anything? no? infinity doesn't exist. You can't just pretend that you've done something an infinite number of times, when you've only written a finite number of symbols. Come on dude this isn't hard to comprehend. I'm guessing you're retarded or something?

>>11595299
>tldr; you can't and you didn't
I've said that you need better arguments than this

>>11594487
You misspelled "correct response".

>>11595299
You being too retarded to understand how to work with infinite values doesn't mean it's impossible to do. Why do you call people who've actually read a fucking book on the subject retarded when you clearly done nothing except eat book covers?

i hate 4chan

>>11591121
because 0.888... never goes past 0.89 so cannot get infinitely close to 0.9

>>11595370
I agree, it's filled with two types of people:
1) retards who think .9...=/=1
2) gullible fish who understand basic analysis, but bite onto every bait thread they see.

>>11594095
If I use your imaginary notation

0.9...95 < 0.9...9...9

>>11591469
>They're all true depending on which axioms you choose to use.
but we obviously aren’t talking any nonstandard here. unless explicitly specified, we operate in the classic real field with the standard notations. go show off somewhere else.

>>11595370
last bastion of free speech.

>>11596221
The fact that this statement is true makes me cry everytime I think about it.

>>11596232
only women children and homosexuals cry

>>11596391
I've been the OP on several threads, so me being a faggot makes total sense lol.

>>11591121
Because only one is at the upper bound of the (completely arbitrary) base chosen to represent 10.

>>11591645
Correct. 0.89999 = 0.9 But this is just the same as 0.999999 = 1

>>11591121
There is no possible number a such that a + 0.999... = 1. This is not the case with the other two

>>11596928
0

>>11596928
This AGAIN? a = 0.000000...1

does the 1 = 0.999... problem mean there are infinitely many tiers of subatomical particles?

>you symbolic equations with infinity
wuh

>>11597198
no

>>11596944
Zero isn't a number. It is the absence of a number

>>11595408
this is a contradiction. 2) is correct but 1) should read “trolls who either know 0.(9)=1 or don’t even care”. same goes for flat earth.

>>11597117
not an element of [math]\mathbb{R}[/math]. try again.

>>11591121
two of them are true.
0.888... and 0.999... both can keep on going forever, and eventually the difference between the next number will be so small that it might as well be the next number. this isn't true for 1.000... since no matter how many zeroes you add it's not going to get it any closer to 1.1

If you accept that
1. All numbers can be represented by infinite decimal expansions in the form of a function f: N->{0,1,2,3,4,5,6,7,8,9, ".", "-"}, (for example, 0.99999... is the function f(0)=0, f(1)=. , f(2)=9, f(3)=9, f(n)=9 for all n>3 . 1 can be represented by g(0)=1, g(1)=., g(2)=g(3)=... = g(n) = 0). A sequence is valid if it has at most one "-" in front and exactly one "." somewhere in the sequence. We assume that every valid representation determines a unique number but not necessarily that every number has a unique decimal representation (for this is what we will want to disprove).
2. You can do arithmetic on numbers in expected ways.
Then you are forced to have
0.9999....=1
Proof:
By comparing each digit, we see that 1 is not less than 0.999....
Now subtract the two numbers (which you can do by assumption 2)
1-0.999... = x >= 0
As every number can be represented by an infinite decimal expansion (assumption 1), there is a sequence of digits for x.
If there are any nonzero digits of x after the dot ".", there is a first nonzero digit and it has a well-defined natural number position N. But clearly that cannot be, since then it would mean that x is bigger than some fraction 1/10^N. Thus every digit of x must be zero and so x itself is zero.
So 1-0.999... = 0
now again, by assumption 2., you can add 0.999... to both sides to get
0.999... = 1
QED

>>11597560
Possible objections you might have:
1. Not all numbers have a sequential decimal expansion of the form f:N->{digits}. Evidently, an expression like 0.0...1 cannot be made into such a sequence because for every position in a natural sequence there are finitely many terms coming before it.
To make sense of an expression like 0.0...1 rigorously, you could instead define it as an ordinal sequence f:(w+1)->digits. w+1 is ordinal obtained by taking the natural numbers and adding one element to the end that is larger than every other element. (Look up ordinals here: https://en.wikipedia.org/wiki/Ordinal_number))
So 0.0....1 could be interpreted as f(0)=0, f(1)=".", f(2)=...=f(n)=0 and f(w)=1.
Now the problem still arises because not only you've gained a lot more numbers in this way, they would not be considered to be numbers by mathematicians because you cannot add, subtract them.
To illustrate this:
What's 10* 0.0...1? Intuitively, we shift the decimal point by 1 (do you have other suggestions?). But that would result in exactly the same representation, hence the same number 0.0...1! And we cannot have that because if 10*0.0...1 = 0.0...1, subtracting 0.0...1 we find 9*0.0...1=0 and so if we assume 0.0...1!=0, we can divide by it to find a blatant contradiction 9=0. So we see that even allowing nonstandard decimal sequences doesn't solve the problem: we need to be able to do arithmetic on numbers in expected ways and assuming 0.99...!=1 always leads us to a contradiction!
2. Not all numbers have decimal representations. In that case, there's not a lot to say here on my part except to ask what do you mean by a number then? Because in all these discussions a prevaling implicit assumption has been that numbers mostly ARE their decimal representations. "What's 0.999..? Obviously it's the number you get by writing 0 and 9999 repeating": there is no notion that it's just a notation that represents some number: it's a number itself.
cont.

>>11597583
<cont.>
So if you believe not all numbers have decimal representations, please provide your own definition of a number. I assume by numbers you don't mean equivalence classes of Cauchy sequences or Dedekind cuts, because in those formulations 0.999... = 1 is a trivial theorem.
3. You cannot do arithmetic in expected ways on all the numbers. In that case, there is nothing left to argue about, it's all just a matter of a definition of what you mean by numbers. For me and most mathematicians, numbers are something you can add, subtract, multiply (although not necessarily divide in themselves, e.g. the integers are not closed under division). Although, appealing to commons sense, in what sense are your numbers "real" if you cannot even add, subtract, multiply them?

That's it for now.

Is no one going to respond to the posts above?

>>11597583
>Not all numbers have decimal representations.
Aren't you understating quite a bit there?
Practically all numbers have no (finite) decimal representation.
That said, the Reals all have an "f" like the guy you were replying to suggests.

>>11597650
Not the guy you were talking with, so I didn't really feel like bumping in.

>>11597762
That was me in all 4 of the posts above.
By decimal representation I meant a function from an ordinal to the set of digits.

>>11597650
Nope. If you want (you)'s, you do it like this:

HEY GUYS, check this out. I have a definite proof that 0.999... < 1, because I have found a number between them.

Just switch to base 16.

Then obviously

0.999... (base 10) < 0.FFF... (base 16) < 1

it's called Geometric Series
https://www.youtube.com/watch?v=cD0QoPYc_LI

0.88888...=8x(Geometric Series for x=0.1)
0.99999....=9x(Geometric Series for x=0.1)
1.00000....=1

>>11597768
>By decimal representation I meant a function from an ordinal to the set of digits.
Well, anything past (and including) [math]\omega[/math] is not a thing in the Reals.

It feels weird how you argue with addition, subtraction, multiplication and division, even though terms like "field" and "group" would facilitate that a lot, but my background is CS, so I may very well be the guy in the wrong here.

DEUS VULT!

>>11597803
Have you not read my posts? I was not talking about the reals. I was talking about ways to make sense of infinite decimal expansions and do arithmetic on them.

>>11597803
dude can you read ? notice this sentence:

>>11597583
>Possible objections you might have:

This is a bad thread.

>>11597779
Your equation are wrong according to your own video it should be
0.88888 = 0.8 [math]\frac{1}{1-x}[/math]
for x=0.1

>>11591887
good troll

>>11591645
The number between 0.999... and 1 is 0.999...

>>11597650
If you want (You)s you have to say something wrong and incredibly dumb
Writing out a careful and considered argument will not get a reaction, which incidentally is the reason why this website is bad.

>>11591121
Decimal system ends at nine nigger

>>11597550
But 0.8... never reaches 0.88898...
What drugs are you on nigger?

>>11594029
>Wrong, .9...=1
That's literally what he said.
>without simplification
You can write the same number multiple ways, e.g. 4/2 or 2 + 2.

>>11601297
*1 + 1

>>11595219
wrong

>>11594060
lol

>>11594104
lel

>>11591121
This is proof positive we need to ban ip addresses which are not from a university or can pass a basic calculus test

>>11602453
I know more math than you

>>11591194
the jews invented 0.999=1

only way to gain is to take. gaining $0.001 overtime from 7 billion people. well you'll be in control at the top in 5 years.

>>11602493
>

did some math

0.001* 7bil = 7mil
times 52 weeks in a year
$364mil* 5 years
$1.8bil in 5 years


This is what happens when 0.999=1 and the jew take out their 0.001 goytax every paycheck. Really makes you think

>>11602505
This is like a leftie trying to shitpost false flag pol but he makes an overkill post and everyone can tell immediately

>>11602459
If you think .9...=/=1, then no you don't.

>>11598774
The arrow means STRICTLY between. Are you arguing that .9...<.9...?

>>11597338
Yes it is you literal retard. Fuck off.

>>11597117
>>11591966
Wrong, you literal retard. The fact that there is a 1 at the END means that there aren't infinite zeroes between them. Don't argue, you're wrong, you misused the notation. Stop being dumb and accept math, and stop being retarded enough to assume that something making sense to YOU is some universal constant for truth.

last thread on page ten --> first thread on page one

>>11605437
Well, there always needs to be a 0.999 =/= 1 thread around, otherwise someone will make another one.
Although the OP is really weak bait this time.

>>11603401
There are infinite numbers in [0,1) and yet something comes after: 1. So you're just wrong

>>11605481
that's true but irrelevant. you can place anything you want after 0.000..., but it will not be a representation of a real number anymore.

>>11591966
>Oh God, another one of THESE threads
>listen up nerds, gonna whip out my dick
>watch me whip
>this shits easy
>.....
>well, I don't want to prove .999...=1
>...I want to prove .000...1 = 0 instead, ya thats it
>.....
>there lim 1/10^n = 0 QED
>proof is right there...
>see, its right there...
>...
>WHOA IS THAT JOHN NASH TALKING TO ERDOS
>I THOUGHT THEY WERE DEAD

why is only 1 of these statements true?

OP is a jew

>>11605574
Pepe memes are not a substitute for understanding analysis.

Why are people still responding to these?

>>11606784
There are frequently """gold nuggets""" hidden in these threads, so it would feel like a shame, if these threads disappeared.
Sure, there is a lot of literal garbage, but you can mentally filter that.
Would probably yield a Fields medal, if you could integrate that filter into 4chanX.

>>11602601
good goy

>>11597583
>What's 10* 0.0...1? Intuitively, we shift the decimal point by 1 (do you have other suggestions?). But that would result in exactly the same representation, hence the same number 0.0...1! And we cannot have that because if 10*0.0...1 = 0.0...1, subtracting 0.0...1 we find 9*0.0...1=0 and so if we assume 0.0...1!=0, we can divide by it to find a blatant contradiction 9=0. So we see that even allowing nonstandard decimal sequences doesn't solve the problem: we need to be able to do arithmetic on numbers in expected ways and assuming 0.99...!=1 always leads us to a contradiction!
https://www.scientificamerican.com/article/strange-but-true-infinity-comes-in-different-sizes/

>>11591966
What then is 0.999... + 0.999... ?
>answer is not 2

>>11607889
The answer is 2

>>11597778
[math] \displaystyle
\begin{align*}
1 = \left (\frac{15}{16} + \frac{1}{16} \right )
&= \text{0x0.F} + \frac{1}{16} \\
= \text{0x0.F} + \left ( \frac{15}{256} + \frac{1}{256} \right )
&= \text{0x0.FF} + \frac{1}{256}\\
= \text{0x0.FF} + \left ( \frac{15}{4096} + \frac{1}{4096} \right )
&= \text{0x0.FFF} + \frac{1}{4096} \\
= \text{0x0.FFF} +\left ( \frac{15}{65536} + \frac{1}{65536} \right )
&= \text{0x0.FFFF} + \frac{1}{65536} \\
&\vdots
\end{align*}
\\ \displaystyle
\Rightarrow \text{0x}0.\overline{\text{F}} = 1
[/math]

>>11594209
retard, talking about it not being aleph_0
You're the idiot who looks at the finger when someone points at the Moon

Imagine being a 1fag

imagine being impressed by finite

>>11607889
>0.999... + 0.999...
= 1.999... = 2

>>11608068
In reality anything sufficiently large is indistinguishable from infinite.
In maths obviously not.

>>11591121
Can see them as limits, first two limits are correct, bottom one is 1, not 1.1

>>11608901
ignore im retarded

>>11591121

3*1/3 = 1
1/3 = 0.3333333...
thus 0.9999999... = 1

>>11593111
i cringed

The real dumb shit is when calc tards start saying dumb nigger shit like [math]\sum_{n=1} \frac{1}{2^n} = \sum_{n=1} \frac{9}{10^n}[/math]

>>11613112
Do you mean

[eqn]{lim}_{m \rightarrow \infty} \sum_{n=1}^{m} \frac{1}{2^n} = {lim}_{m \rightarrow \infty} \sum_{n=1}^{m} \frac{9}{10^n}[/eqn]
That would be correct, but there is no [math]m \in \mathbb{N}[/math] such that would equal.

>>11613112
That's true though.

>>11613112
1st is binary numbers, 2nd is decimal numbers
0.500000000 1*2^-1
0.250000000 1*2^-2
0.125000000 1*2^-3
0.062500000 1*2^-4
0.031250000 1*2^-5
0.015625000 1*2^-6
0.007812500 1*2^-7
0.003906250 1*2^-8
0.001953125 1*2^-9
0.000976563 1*2^-10
0.000488281 1*2^-11
0.000244141 1*2^-12
0.000122070 1*2^-13
0.000061035 1*2^-14
0.000030518 1*2^-15
0.000015259 1*2^-16
0.000007629 1*2^-17
0.000003815 1*2^-18
0.000001907 1*2^-19
0.000000954 1*2^-20
0.000000477 1*2^-21
0.000000238 1*2^-22
0.000000119 1*2^-23
0.000000060 1*2^-24
0.000000030 1*2^-25
0.000000015 1*2^-26
0.000000007 1*2^-27
0.000000004 1*2^-28
0.000000002 1*2^-29
0.000000001 1*2^-30
0.000000000 1*2^-31
0.000000000 1*2^-32
0.000000000 1*2^-33

1.000000000